Randomized Complete Block with One Factor

This example illustrates the use of PROC ANOVA in analyzing a randomized complete block design. Researchers are interested
in whether three treatments have different effects on the yield and worth of a particular crop. They believe that the experimental
units are not homogeneous. So, a blocking factor is introduced that allows the experimental units to be homogeneous within
each block. The three treatments are then randomly assigned within each block.

The variables Yield and Worth are continuous response variables, and the variables Block and Treatment are the classification variables. Because the data for the analysis are balanced, you can use PROC ANOVA to run the analysis.

The “Class Level Information” table lists the number of levels and their values for all effects specified in the CLASS statement. The number of observations in the data set are also displayed. Use this information to make sure that the data
have been read correctly.

The overall ANOVA table for Yield in Figure 26.6 appears first in the output because it is the first response variable listed on the left side in the MODEL statement.

Figure 26.6: Overall ANOVA Table for Yield

Randomized Complete Block

The ANOVA Procedure

Dependent Variable: Yield

Source

DF

Sum of Squares

Mean Square

F Value

Pr > F

Model

4

225.2777778

56.3194444

8.94

0.0283

Error

4

25.1911111

6.2977778

Corrected Total

8

250.4688889

R-Square

Coeff Var

Root MSE

Yield Mean

0.899424

6.840047

2.509537

36.68889

The overall F statistic is significant , indicating that the model as a whole accounts for a significant portion of the variation in Yield and that you can proceed to evaluate the tests of effects.

The degrees of freedom (DF) are used to ensure correctness of the data and model. The Corrected Total degrees of freedom are
one less than the total number of observations in the data set; in this case, 9 – 1 = 8. The Model degrees of freedom for
a randomized complete block are , where b = number of block levels and t = number of treatment levels. In this case, this formula leads to model degrees of freedom.

Several simple statistics follow the ANOVA table. The R-Square indicates that the model accounts for nearly 90% of the variation
in the variable Yield. The coefficient of variation (C.V.) is listed along with the Root MSE and the mean of the dependent variable. The Root MSE
is an estimate of the standard deviation of the dependent variable. The C.V. is a unitless measure of variability.

The tests of the effects shown in Figure 26.7 are displayed after the simple statistics.

Figure 26.7: Tests of Effects for Yield

Source

DF

Anova SS

Mean Square

F Value

Pr > F

Block

2

98.1755556

49.0877778

7.79

0.0417

Treatment

2

127.1022222

63.5511111

10.09

0.0274

For Yield, both the Block and Treatment effects are significant and , respectively) at the 95% level. From this you can conclude that blocking is useful for this variable and that some contrast
between the treatment means is significantly different from zero.

Figure 26.8 shows the ANOVA table, simple statistics, and tests of effects for the variable Worth.

Figure 26.8: ANOVA Table for Worth

Randomized Complete Block

The ANOVA Procedure

Dependent Variable: Worth

Source

DF

Sum of Squares

Mean Square

F Value

Pr > F

Model

4

1247.333333

311.833333

8.28

0.0323

Error

4

150.666667

37.666667

Corrected Total

8

1398.000000

R-Square

Coeff Var

Root MSE

Worth Mean

0.892227

4.949450

6.137318

124.0000

Source

DF

Anova SS

Mean Square

F Value

Pr > F

Block

2

354.6666667

177.3333333

4.71

0.0889

Treatment

2

892.6666667

446.3333333

11.85

0.0209

The overall F test is significant at the 95% level for the variable Worth. The Block effect is not significant at the 0.05 level but is significant at the 0.10 confidence level . Generally, the usefulness of blocking should be determined before the analysis. However, since there are two dependent variables
of interest, and Block is significant for one of them (Yield), blocking appears to be generally useful. For Worth, as with Yield, the effect of Treatment is significant .

Issuing the following command produces the Treatment means.

means Treatment;
run;

Figure 26.9 displays the treatment means and their standard deviations for both dependent variables.